Finite Group Databasebeta

← #78986

Subgroups — #78986

Order 384.

724 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5964C211yesmaximal, characteristic, complemented
2#5822C211yesmaximal, characteristic, complemented
3#5748C211yesmaximal, characteristic, complemented
4#5477C211yesmaximal, characteristic, complemented
5C2×C3×#104C211yesmaximal, characteristic, complemented
6#5132C211yesmaximal, characteristic, complemented
7C2×#805C211yesmaximal, characteristic, complemented
8#4827C212yesmaximal, complemented
9#4826C212yesmaximal, complemented
10#4825C212yesmaximal, complemented
11#4822C212yesmaximal, complemented
12#3018—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14#1010—21no—
15#1010C2212yes—
16#1009—21no—
17#1009C2212yes—
18D3×#41—22no—
19D3×#41—22no—
20#1007—21no—
21#1004—21no—
22#1004—22no—
23C2×#235—22no—
24C2×#235C2211yescharacteristic
25C2×#235C2211yescharacteristic
26C2×#235C2211yescharacteristic, complemented
27C2×C3×M16C2211yescharacteristic, complemented
28C2×C3×C42C2211yescharacteristic, complemented
29#951C2212yescomplemented
30#950—22no—
31#949C2212yescomplemented
32#942—21no—
33#932—21no—
34#930—22no—
35C2×#208C2211yescharacteristic, complemented
36#909—21no—
37#909C2212yescomplemented
38#908—21no—
39#908C2212yescomplemented
40#904—22no—
41#875C2211yescharacteristic, complemented
42#875C2211yescharacteristic, complemented
43#874C2211yescharacteristic, complemented
44C4×D12—22no—
45#869C2211yescharacteristic, complemented
46C3×#104C2212yescomplemented
47C3×#104C2212yescomplemented
48#824C2212yescomplemented
49#823C2212yescomplemented
50#805C2212yescomplemented
51#805C2212yescomplemented
52#585—31no—
53#575—31no—
54#575—31no—
55#532—31no—
56#456—32no—
57#455—32no—
58#454—32no—
59#453—32no—
60C2×#104—31no—
61C2×#104D311yescharacteristic, complemented
62#413—31no—
63C3×#41—21no—
64C3×#41—21no—
65C3×#41D412yescomplemented
66C3×#41C2312yes—
67C2×C3×Q8D411yescharacteristic, complemented
68C2×C3×D4—21no—
69C2×C3×D4D411yescharacteristic, complemented
70C22×C3×C4—21no—
71C22×C3×C4—21no—
72C22×C3×C4C2311yescharacteristic
73C2×#67—24no—
74C2×#67—21no—
75C2×#67—21no—
76C2×#67D412yes—
77#239—22no—
78#239—22no—
79D3×Q8—22no—
80D3×Q8—22no—
81#237—24no—
82#237—22no—
83#237—22no—
84#237—24no—
85D3×D4—24no—
86D3×D4—22no—
87D3×D4—24no—
88D3×D4—22no—
89#235—24no—
90#235—24no—
91#235—21no—
92#235—21no—
93#235—21no—
94#235—21no—
95#235—21no—
96#235—21no—
97#235—22no—
98#235—21no—
99#235—21no—
100#235—21no—
101#235—21no—
102#235—22no—
103#235D412yescomplemented
104#235C2312yes—
105C2×D12—22no—
106C2×D12—22no—
107C2×D12D411yescharacteristic, complemented
108C2×D12C2311yescharacteristic
109C2×D12C2311yescharacteristic
110C2×C4×D3—24no—
111C2×C4×D3—21no—
112C2×C4×D3—21no—
113C2×C4×D3D412yes—
114C2×Dic6—22no—
115C2×Dic6D411yescharacteristic, complemented
116C2×Dic6C2311yescharacteristic
117C2×Dic6C2311yescharacteristic
118C3×M16—21no—
119C3×M16C2312yes—
120C2×C3×C8—21no—
121C3×C42—21no—
122C3×C42C2311yescharacteristic
123C3×C42C2311yescharacteristic
124#216—22no—
125#215—22no—
126#214—22no—
127#213—22no—
128(C2×D3)⋊C4—22no—
129(C2×D3)⋊C4—22no—
130C4⋊Dic3—22no—
131#208—21no—
132#208C2312yes—
133C2×C3⋊C8—21no—
134Dic12—22no—
135D24—22no—
136#204—22no—
137#204—22no—
1382−1+4—32no—
1392−1+4—61no—
1402−1+4—61no—
141H(5,𝔽2)—32no—
142H(5,𝔽2)—62no—
143H(5,𝔽2)—61no—
144H(5,𝔽2)—61no—
145C2×#41—31no—
146C2×#41—31no—
147C2×#41—31no—
148C2×#41—62no—
149C2×#41—62no—
150C2×#41—62no—
151C2×#41C2×D311yescharacteristic
152#137—32no—
153#137—32no—
154#137—61no—
155#137—61no—
156AGL(1,ℤ/8)—32no—
157AGL(1,ℤ/8)—32no—
158AGL(1,ℤ/8)—61no—
159AGL(1,ℤ/8)—61no—
160#135—62no—
161#135—62no—
162C2×M16—31no—
163C2×M16C2×D311yescharacteristic, complemented
164C4⋊Q8—31no—
165#127—31no—
166#124—31no—
167#124—31no—
168#121—62no—
169C4×D4—62no—
170C2×C42C2×D311yescharacteristic, complemented
171#104—32no—
172#104—32no—
173#104C2×D312yescomplemented
174#104C2×D312yescomplemented
175#101—32no—
176#100—32no—
177C23×C3—21no—
178C23×C3C2×D411yescharacteristic
179C22×D3—24no—
180C22×D3—21no—
181C22×D3—21no—
182C22×D3C2×D412yes—
183C3×Q8—21no—
184C3×Q8C2×D412yes—
185C3×D4—22no—
186C3×D4—22no—
187C3×D4—21no—
188C3×D4C2×D412yes—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4—21no—
194C2×C3×C4—42no—
195C2×C3×C4C2×D411yescharacteristic
196C2×C3×C4C2×D411yescharacteristic
197C2×C3×C4C2×D411yescharacteristic
198C2×C3×C4C2×D411yescharacteristic
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2411yescharacteristic
201#67—24no—
202#67—24no—
203#67—22no—
204#67—22no—
205#67—22no—
206#67—22no—
207#67—41no—
208#67—41no—
209C2×Dic3—24no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3C2×D412yes—
213D12—24no—
214D12—21no—
215D12—21no—
216D12—21no—
217D12—21no—
218D12—21no—
219D12—44no—
220D12C2×D412yes—
221C4×D3—24no—
222C4×D3—24no—
223C4×D3—22no—
224C4×D3—22no—
225C4×D3—22no—
226C4×D3—22no—
227C4×D3—41no—
228C4×D3—41no—
229Dic6—24no—
230Dic6—21no—
231Dic6—21no—
232Dic6—21no—
233Dic6—21no—
234Dic6—21no—
235Dic6C2×D412yes—
236C3×C8—22no—
237C3⋊C8—22no—
238#41—21no—
239#41—21no—
240#41—32no—
241#41—32no—
242#41—62no—
243#41—62no—
244#41—64no—
245#41—64no—
246#41—64no—
247#41—64no—
248#41—62no—
249#41—62no—
250#41—62no—
251#41—62no—
252#41—61no—
253#41—61no—
254#41—61no—
255#41—61no—
256#41—61no—
257#41—61no—
258#41—61no—
259#41—61no—
260#41—61no—
261#41—61no—
262#41D1212yescomplemented
263#41C22×D312yes—
264C2×Q8—31no—
265C2×Q8—31no—
266C2×Q8—31no—
267C2×Q8—62no—
268C2×Q8—62no—
269C2×Q8—62no—
270C2×Q8D1211yescharacteristic, complemented
271C2×D4—21no—
272C2×D4—32no—
273C2×D4—31no—
274C2×D4—31no—
275C2×D4—31no—
276C2×D4—62no—
277C2×D4—62no—
278C2×D4—64no—
279C2×D4—64no—
280C2×D4—62no—
281C2×D4—64no—
282C2×D4—61no—
283C2×D4—62no—
284C2×D4—61no—
285C2×D4D1211yescharacteristic, complemented
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—32no—
289C22×C4—64no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4C22×D311yescharacteristic
293Q16—62no—
294Q16—62no—
295SD16—62no—
296SD16—62no—
297SD16—62no—
298SD16—62no—
299D8—62no—
300D8—62no—
301M16—21no—
302M16—32no—
303M16—61no—
304M16C22×D312yes—
305C2×C8—21no—
306C2×C8—61no—
307C4⋊C4—62no—
308C22⋊C4—62no—
309C22⋊C4—62no—
310C42—21no—
311C42C22×D311yescharacteristic
312C42C22×D311yescharacteristic
313C22×C3—22no—
314C22×C3—41no—
315C22×C3—81no—
316C22×C3#12011yescharacteristic
317C22×C3#12011yescharacteristic
318C22×C3C22×D411yescharacteristic
319C2×D3—22no—
320C2×D3—22no—
321C2×D3—42no—
322C2×D3—42no—
323C2×D3—41no—
324C2×D3—81no—
325C2×D3—84no—
326C3×C4—22no—
327C3×C4—22no—
328C3×C4—41no—
329C3×C4#12011yescharacteristic
330C3×C4#12011yescharacteristic
331C3×C4C22×D411yescharacteristic
332C3×C4C22×D411yescharacteristic
333Dic3—22no—
334Dic3—22no—
335Dic3—41no—
336C23—21no—
337C23—32no—
338C23—64no—
339C23—61no—
340C23—61no—
341C23D3×D411yescharacteristic
342Q8—21no—
343Q8—32no—
344Q8—64no—
345Q8—61no—
346Q8—61no—
347Q8—61no—
348Q8—61no—
349Q8—61no—
350Q8C2×D1212yes—
351D4—22no—
352D4—22no—
353D4—21no—
354D4—32no—
355D4—64no—
356D4—64no—
357D4—64no—
358D4—62no—
359D4—62no—
360D4—62no—
361D4—62no—
362D4—61no—
363D4—61no—
364D4—61no—
365D4—61no—
366D4—61no—
367D4—124no—
368D4—121no—
369D4—121no—
370D4C2×D1212yes—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—32no—
377C2×C4—42no—
378C2×C4—64no—
379C2×C4—64no—
380C2×C4—64no—
381C2×C4—61no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—61no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4C2×D1211yescharacteristic
390C2×C4C2×D1211yescharacteristic
391C2×C4D3×D411yescharacteristic
392C2×C4D3×D411yescharacteristic
393C2×C4D3×D411yescharacteristic
394C2×C4C23×D311yescharacteristic
395C8—22no—
396C8—62no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—42no—
401C2×C3C2×#12011yescharacteristic
402D3—42no—
403D3—42no—
404D3—81no—
405C22—22no—
406C22—41no—
407C22—62no—
408C22—62no—
409C22—81no—
410C22—122no—
411C22—122no—
412C22—121no—
413C22—241no—
414C22—244no—
415C22#88211yescharacteristic
416C22#88211yescharacteristic
417C22C2×D3×D411yescharacteristic
418C4—22no—
419C4—22no—
420C4—41no—
421C4—62no—
422C4—62no—
423C4—121no—
424C4#88211yescharacteristic
425C4#88211yescharacteristic
426C4C22×D1211yescharacteristic
427C4C2×D3×D411yescharacteristic
428C3#301811yescharacteristic, complemented, Sylow
429C2—21no—
430C2—21no—
431C2—21no—
432C2—42no—
433C2—122no—
434C2—122no—
435C2—241no—
436C2C2×#88211yescharacteristic
subgroupG/H
centerC4#882
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#104C2
socleC2×C3C2×#120
hypercenterC2×#104D3
hypocenterC3#3018
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#104

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #5964C2semidirect
2. #5822C2semidirect
3. #5748C2semidirect
4. #5477C2semidirect
5. C2×C3×#104C2semidirect
6. #5132C2semidirect
7. C2×#805C2semidirect
8. #4827C2semidirect
9. #4826C2semidirect
10. #4825C2semidirect
11. #4822C2semidirect
26. C2×#235C22semidirect
27. C2×C3×M16C22semidirect
28. C2×C3×C42C22semidirect
29. #951C22semidirect
31. #949C22semidirect
35. C2×#208C22semidirect
37. #909C22semidirect
39. #908C22semidirect
41. #875C22semidirect
42. #875C22semidirect
43. #874C22semidirect
45. #869C22semidirect
46. C3×#104C22semidirect
47. C3×#104C22semidirect
48. #824C22semidirect
49. #823C22semidirect
50. #805C22semidirect
51. #805C22semidirect
61. C2×#104D3semidirect
65. C3×#41D4semidirect
67. C2×C3×Q8D4semidirect
69. C2×C3×D4D4semidirect
103. #235D4semidirect
107. C2×D12D4semidirect
115. C2×Dic6D4semidirect
163. C2×M16C2×D3semidirect
170. C2×C42C2×D3semidirect
173. #104C2×D3semidirect
174. #104C2×D3semidirect
262. #41D12semidirect
270. C2×Q8D12semidirect
285. C2×D4D12semidirect
428. C3#3018semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#59642
#58222
#57482
#54772
C2×C3×#1042
#51322
C2×#8052
#48272
#48262
#48252
#48222
#30183